o rotational transitions o vibrational transitions o ... o rotational transitions o vibrational...
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o Rotational transitions
o Vibrational transitions
o Electronic transitions
o Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated.
o This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj):
o Involves the following assumptions:
o Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed.
o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons.
"molecule (ˆ r i, ˆ R j ) ="electrons( ˆ r i, ˆ R j )"nuclei( ˆ R j )
o Electronic transitions: UV-visible
o Vibrational transitions: IR
o Rotational transitions: Radio
Electronic Vibrational Rotational
o Must first consider molecular moment of inertia:
o At right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”.
o Generally specified about three axes: Ia, Ib, Ic.
o For linear molecules, the moment of inertia about the internuclear axis is zero.
o See Physical Chemistry by Atkins.
I = miri 2
o Rotation of molecules are considered to be rigid rotors.
o Rigid rotors can be classified into four types:
o Spherical rotors: have equal moments of inertia (e.g., CH4, SF6).
o Symmetric rotors: have two equal moments of inertial (e.g., NH3).
o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).
o Asymmetric rotors: have three different moments of inertia (e.g., H2O).
o The classical expression for the energy of a rotating body is:
where !a is the angular velocity in radians/sec.
o For rotation about three axes:
o In terms of angular momentum (J = I!):
o We know from QM that AM is quantized:
o Therefore, , J = 0, 1, 2, …
Ea =1/2Ia"a 2
E =1/2Ia"a 2 +1/2Ib"b
2 +1/2Ic"c 2
E = Ja 2
2Ia + Jb 2
2Ib + Jc 2
J = J(J +1)!2
EJ = J(J +1)! 2I
, J = 0, 1, 2, …
o Last equation gives a ladder of energy levels.
o Normally expressed in terms of the rotational constant, which is defined by:
o Therefore, in terms of a rotational term:
o The separation between adjacent levels is therefore
F(J) - F(J-1) = 2BJ
o As B decreases with increasing I =>large molecules have closely spaced energy levels.
hcB = ! 2
2I => B = !
F(J) = BJ(J +1)
o Transitions are only allowed according to selection rule for angular momentum:
"J = ±1
o Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor.
o Note, the intensity of each line reflects the populations of the initial level in each case.
o Consider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement
(F = -kx). Potential energy is therefore
V = 1/2 kx2
o Can write the corresponding Schrodinger equation as
o The SE results in allowed energies
2µ d2" dx 2
+ [E #V ]" = 0
2µ d2" dx 2
+ [E #1/2kx 2]" = 0
µ = m1m2 m1 + m2
Ev = (v +1/2)!"
" = k µ
$ % &
v = 0, 1, 2, …
o The vibrational terms of a molecule can therefore be given by
o Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond.
o A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k.
G(v) = (v +1/2) ˜ v
˜ v = 1 2"c
$ % &
o The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations.
o Transition occur for "v = ±1
o This potential does not apply to energies close to dissociation energy.
o In fact, parabolic potential does not allow molecular dissociation.
o Therefore more consider anharmonic oscillator.
o A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations.
o At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit.
o Must therefore use a asymmetric potential. E.g., The Morse potential:
where De is the depth of the potential minimum and
V = hcDe 1" e "a(R"Re )( )
a = µ" 2
o The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels:
where xe is the anharmonicity constant:
o The second term in the expression for G increases with v => levels converge at high quantum numbers.
o The number of vibrational levels for a Morse oscillator is finite:
v = 0, 1, 2, …, vmax
G(v) = (v +1/2) ˜ v " ( ˜ v +1/2)2 xe ˜ v
xe = a2! 2µ"
o Molecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J)
o Selection rules obtained by combining rotational selection rule !J = ±1 with vibrational rule !v = ±1.
o When vibrational transitions of the form v + 1 ! v occurs, !J = ±1.
o Transitions with !J = -1 are called the P branch:
o Transitions with !J = +1 are called the R branch:
o Q branch are all transitions with !J = 0
S(v,J) = (v +1/2) ˜ v + BJ(J +1)
˜ v P (J) = S(v +1,J "1) " S(v,J) = ˜ v " 2BJ
˜ v R (J) = S(v +1,J +1) " S(v,J) = ˜ v + 2B(J +1)
o Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV).
o Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy).
o Electronic transitions occur between molecular orbitals.
o Must adhere to angular momentum selection rules.
o Molecular orbitals are labeled, ", #, $, … (analogous to S, P, D, … for atoms)
o For atoms, L = 0 => S, L = 1 => P o For molecules, % = 0 => ", % = 1 => #
o Selection rules are thus
$% = 0, ±1, $S = 0, $"=0, $& = 0, ±1
o Where & = % + " is the total angular momentum (orbit and spin).